Now we're getting somewhere interesting and worthy of debate. Not whether a name is offensive, but the general topic of changing the names of sports teams and what effect that has.
This, I find, is a cultural matter. As an outsider looking in, the United States is no stranger to 'franchising' teams and has had a long tradition of teams not only changing names, but moving towns too.
Detroit Gems moved to Minneapolis and became the Lakers, before relocating again to Los Angeles. (NBA)
Charlotte Hornets moved to New Orleans becoming the New Orleans Hornets. Charlotte gained a new franchise called the Bobcats. New Orleans Hornets renamed themselves the New Orleans Pelicans, so Charlotte Bobcats became Charlotte Hornets once again. (NBA)
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Chicago Cardinals moved to St. Louis, before moving again to Phoenix, then Glendale, AZ to become the Arizona Cardinals. (NFL)
Cleveland Browns moved to Baltimore, becoming the Ravens. Cleveland was awarded a new Browns franchise, and is considered a 'true' continuation of the original Cleveland Browns. (NFL)
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Boston Braves moved to Milwaukee, before finally settling in Atlanta. (MLB)
Brooklyn Dodgers moved to San Francisco, while New York Giants moved to San Fransisco. (MLB)
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San Jose Earthquakes moved to Houston, becoming the Dynamo. San Jose was later awarded a new Earthquakes franchise. (MLS)
It goes on - Wiki
How, as a fan of a relocated team, does one react? Particularly with the Cleveland Browns issue, as this is one of the most controversial. The 'new' 1999 Cleveland Browns are considered a continuation of the Browns who last played in Cleveland in 1996 despite everything to do with that team, the players, the manager, the coaching staff, all of them relocating to Baltimore in time for 1997. I can understand why fans look at at as the same team, but looking at the cold, hard facts it simply isn't true.
In the United Kingdom the mere thought of moving a team to a new city or changing its name is considered treason and will genuinely start mass, sometimes violent, protests. It wasn't always the case; in the embryonic years of professional sports as a discipline, soccer teams frequently changed names (Newton Heath Lancashire & Yorkshire Railway -> Manchester United) (Dial Square -> Royal Arsenal -> Woolwich Arsenal -> Arsenal) (Old King's Scholars -> Chester -> Chester City) and it is still not uncommon for teams to move to a different stadium
within the same existing geographical location, but actual relocation and renaming in the modern age is unheard of.
Wimbledon FC moved from their home in Merton, London to Milton Keynes some c.40 miles away in 2003 and the following year were renamed Milton Keynes Dons. A splinter group of fans started their own AFC Wimbledon at the bottom of the league system and despite all players, staff, managers moving to Milton Keynes in 2003, AFC Wimbledon are considered the successor club and even have the 'old' Wimbledon's trophies. MK Dons retained their positions in the league system and national cup competitions, yet are not considered the same team which played in 2002. I am not saying I agree with the move, but in terms of continuity, they are the same team whether one likes it or not.
Should the Washington Redskins ultimately change their name, for
whatever reason in the end, fans should be thankful that the team is still in their city because it's not at all rare for teams over there to suddenly uproot and leave fans with nothing. Lest we forget, it was when the Boston Redskins moved to the capital that we got this team in the first place.
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tl;dr - Never mind whether the name is racist or not, what would a name change do to the fans either way? And be thankful the team is still in DC and not somewhere else.